Bounded prime gaps in short intervals

TL;DR

This paper extends Zhang's and Pintz's work on bounded prime gaps to short intervals, providing a lower bound on prime pairs within a specific range, and discusses potential improvements with sharper variants.

Contribution

It generalizes existing results on prime gaps to short intervals and analyzes Zhang's proof to establish new bounds, also suggesting how sharper theorems could improve these bounds.

Findings

Lower bound for prime pairs in short intervals established

Analysis of Zhang's proof methodology provided

Potential for improved bounds with sharper variants

Abstract

We generalise Zhang's and Pintz recent results on bounded prime gaps to give a lower bound for the the number of prime pairs bounded by 6*10^7 in the short interval $[x,x+x (\log x)^{-A}]$. Our result follows only by analysing Zhang's proof of Theorem 1, but we also explain how a sharper variant of Zhang's Theorem 2 would imply the same result for shorter intervals.